Properties
Reductive Lie algebras are a generalization of semisimple Lie algebras, and share many properties with them: many properties of semisimple Lie algebras depend only on the fact that they are reductive. Notably, the unitarian trick of Hermann Weyl works for reductive Lie algebras.
The associated reductive Lie groups are of significant interest: the Langlands program is based on the premise that what is done for one reductive Lie group should be done for all.
The intersection of reductive Lie algebras and solvable Lie algebras is exactly abelian Lie algebras (contrast with the intersection of semisimple and solvable Lie algebras being trivial).
Read more about this topic: Reductive Lie Algebra
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)